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Abstract

We demonstrate that tunable attractive (bonding) and repulsive (anti-bonding) forces can arise in highly asymmetric structures coupled to external radiation, a consequence of the bonding/anti-bonding level repulsion of guided-wave resonances that was first predicted in symmetric systems. Our focus is a geometry consisting of a photonic-crystal (holey) membrane suspended above an unpatterned layered substrate, supporting planar waveguide modes that can couple via the periodic modulation of the holey membrane. Asymmetric geometries have a clear advantage in ease of fabrication and experimental characterization compared to symmetric double-membrane structures. We show that the asymmetry can also lead to unusual behavior in the force magnitudes of a bonding/antibonding pair as the membrane separation changes, including nonmonotonic dependences on the separation. We propose a computational method that obtains the entire force spectrum via a single time-domain simulation, by Fourier-transforming the response to a short pulse and thereby obtaining the frequency-dependent stress tensor. We point out that by operating with two, instead of a single frequency, these evanescent forces can be exploited to tune the spring constant of the membrane without changing its equilibrium separation.

Figures (6)

Schematic of single-membrane (asymmetric) structure: a photonic-crystal (holey) membrane (thickness h1 = 0.2a) consisting of a square-lattice of air holes (radius R = 0.2a) on silicon is suspended (separation s) on top of an unpatterned (homogeneous) silicon slab (thickness h2 = 0.2a) sitting on top of a semi-infinite silica substrate. Light is incident on the membrane from the normal direction (top).

(Left:) Resonance frequency ω (units of 2πc/a) a function of separation s (units of membrane period a), for both the single-membrane (asymmetric) structure of Fig. 1 (solid lines) as well as the double-membrane (symmetric) structure of Ref. [24] (dashed lines). The insets show the electric field component Ex in the x–z plane (y = 0) near ω at a particular s = 0.3a. In the symmetric case, the attractive and repulsive modes are in-phase and out of phase, respectively, as expected. (Right:) Resonant (peak) force Fc/P (units of incident power P/c), at the resonant frequencies ω plotted on the left figure, as a function of s. The bottom inset shows the broad-bandwidth force spectrum of the asymmetric structure at a particular s = 0.2a, showing both the bonding (F > 0) and antibonding (F < 0) resonances. The inset also denotes what is meant by resonance frequency ω and peak force F.

(Left:) Optical force Fc/P on the single-membrane structure of Fig. 1, as a function of the frequency ω of incident light of power P, for various separations s. The insets show typical Ex field patterns (in the x–z plane, at y = 0) for both the attractive (left) and repulsive (right) resonances. (Right:) Optical force Fc/P as a function of separation s for incident light input at various frequencies ω ∈ [0.48, 0.5] (2πc/a). The bottom inset shows Fc/P for light input over a lower frequency range ω ∈ [0.41, 0.43] 2πc/a. The force versus s plot was obtained by fitting the force spectrum obtained via FDTD at a few s to a sum of Lorentzian resonances, and then interpolating the resulting Lorentzian parameters over a denser s range.

(Left:) Optical force Fc/P as a function of separation s, for light incident at two frequencies ω+ (varied) and ω− = 0.495 (2πc/a), with corresponding power P+ and P−, respectively. Dashed lines show the force for P− = 0. (Right:) Absolute value of optical spring constant |κo| (units of P/ca) as a function of frequency ω+. Dashed and solid lines correspond to negative (unstable) and positive (stable) values of κo, plotted for different values of η = P−/P+. Both Fc/P and κo are normalized against the total input power P = P+ + P−.

(Left:) Optical force Fc/P as a function of frequency ω for light of power P incident on the single-membrane structure of Fig. 1, for various separations s. The bottom inset shows the force (solid lines) and reflection (dashed line) of the same geometry but for h2 = 0 and s = 0.3a. (Right:) Corresponding reflection spectrum R as a function of ω. The open circles indicate frequencies for which there exist force minima or maxima. The insets show the electric field component Ex in the x–z plane (y = 0) at a particular s = 0.2a, and at the indicated frequency points ω = 0.64(2πc/a) (left) and ω = 0.681(2πc/a) (right).

Schematic of system consisting of two multilayer objects [labeled as (1) and (2)] separated by a distance s. A two-dimensional cross-section for the particular case of two quarter-wave stack mirrors with a defect (yellow) is shown on the right.